Model Rainfall Extremes on an Australian Wide Scale
Model Rainfall Extremes on an Australian Wide Scale
Distance for clustering is based on extremal dependence.
Calculate our distance \(d(x, x+h)\) using the F-madogram (Cooley 2006):
\[ d(x, x+h) = \dfrac{1}{2} \mathbb{E} \left| \left[ F(Z(x)) - F(Z(x+h)) \right| \right] \] where
\(F\) is the empirical distribution function of
\(Z(x)\) the annual maximum rainfall at location
\(x\) .
Pros: Non-parametric, fast and only use the raw data.
Can write our F-madogram as a function of the extremal coefficient, \(\theta(h)\) \[ d(x,x + h) = \dfrac{\theta(h) - 1}{2(\theta(h) + 1)}, \]
where \(\theta(h)\) is a measure of extremal dependence,
\[ \mathbb{P}\left( Z(x) \leq z, Z(x + h) \leq z \right) = \left[\mathbb{P}(Z(x)\leq z) \right]^{\theta(h)}, \quad x,h \in \mathbb{R}^2 \] where
\(Z\) is unit Frechet.
(For full details refer to Bernard et al. 2013.)
Local level:
+ Fit a Max-stable process
+ Natural extension of univariate extreme value theory to extremes in continuous space with dependence
This work has been supported by the Australian Research Council through the Laureate Fellowship FL130100039, CSIRO and ACEMS.